would like to be able to set this as an option on the category page, so that categories that currently have no products for sale are not shown on the website.
 
This should also apply to parent categories where all down level child categories have no products.
 
this will reduce the administrative overhead of having to hide and re-shuffle categories where there are currently no products for sale for a particular category

This would be a nice feature.

That makes such sense, Hunny
At present we need to manually un-check the category but then we must remember to check it live again when a product is added - easily missed.

I agree, this is a no-brainer to me. If there are no products, the category should not appear.